Nearby Cycle Sheaves for Stable Polar Representations

Let G|V, G connected, reductive over C, be a stable polar representation in the sense of [DK], satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. This provides a partial generalization of the results of [Gr1] and [GVX1]. Our main intended application is to the theory of character sheaves for graded Lie algebras over C.